Complex Bifurcation Structures in the Hindmarsh–Rose Neuron Model

International Journal of Bifurcation and Chaos, Vol. 17, No. 9 (2007) 3071–3083 c World Scientiﬁc Publishing Company

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

J. M. GONZALEZ-MIRANDA´ Departamento de F´ısica Fundamental, Universidad de Barcelona, Avenida Diagonal 647, Barcelona 08028, Spain

Received June 12, 2006; Revised October 25, 2006

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

Keywords: Neuronal dynamics; neuronal coding; deterministic chaos; bifurcation diagram.

1. Introduction 1961] has proven to give a good qualitative descrip- There are two main problems in neuroscience whose tion of the dynamics of the membrane potential of solution is linked to the research in nonlinear a single neuron, which is the most relevant experi- dynamics and chaos. One is the problem of inte- mental output of the neuron dynamics. Because of grated behavior of the nervous system [Varela et al., the enormous development that the ﬁeld of dynam- 2001], which deals with the understanding of the ical systems and chaos has undergone in the last mechanisms that allow the diﬀerent parts and units thirty years [Hirsch et al., 2004; Ott, 2002], the of the nervous system to work together, and appears study of meaningful mathematical systems like the related to the synchronization of nonlinear dynami- Hindmarsh–Rose model have acquired new interest cal oscillators. The other is the neural coding prob- [Holmes, 2005] because of the new information that can now be obtained from them.

Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com lem [Sejnowski, 1995; Abeles, 2004], which is aimed to know how the neurons encode the information In particular, it has been found recently that that they transmit and exchange in the working the dynamics of this system, when studied as a of the nervous system, and is related to both, the function of significant control parameters, presents knowledge of the different types of dynamics avail- two relevant features: (i) continuous interior cri- able to nonlinear dynamical systems and to chaos sis [González-Miranda, 2003], which are abrupt synchronization. changes in the nature of the dynamics of the sys- The theoretical study of such problems, in its tem attractor, and (ii) block structured dynamics by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. most basic aspects, partly relies on mathematical [González-Miranda, 2005], which is a complex bifur- models of the electrophysics of a single neuron, cation structure where the dynamics available to the most of them developed from the pioneering work system are grouped in blocks of alike periodicity. by Hodgkin and Huxley [1952]. Among them, the The first of these items provides a potential mech- model by Hindmarsh and Rose [1984], which is the anism for rapid responses in the nervous system result of a mapping of the Hodgkin–Huxley model by switching between different dynamical behav- to a relatively simple nonlinear oscillator [Fitzhugh, iors. The second provides structures that appear as

3071 3072 J. M. Gonz´alez-Miranda

possible basic elements to encode messages in the 1.5 apparently random trains of action potentials that x run along the axon of an activated neuron. 0 -1.5 These two phenomena have received detailed study at particular restricted values of the 1.5 Hindmarsh–Rose model parameters. The aim of the x 0 present paper is to report a comprehensive study of -1.5 the manifestation of these phenomena for a wide 1.5 range of system parameters; that is, to provide x 0 a phase diagram of the Hindmarsh–Rose model, -1.5 which gives a global view of the availability of these 1.5 and other dynamical behaviors to this system. x This article is arranged as follows. After this 0 introduction, in Sec. 2, we give a basic description -1.5 of the complexities in the bifurcation diagrams of 1.5 the Hindmarsh–Rose neuron model. In Sec. 3, we x 0 present the results of a linear stability analysis in a -1.5 wide and significant region of the parameter space. 1.5 In Sec. 4, we analyze stability from the nonlinear x 0 point of view by means of the use of Lyapunov -1.5 exponents, and go further in the nonlinear analy- 1.5 sis giving a global view of the bifurcation diagrams. x In Sec. 5, we provide evidence for the structural sta- 0 bility of the above results, and finally, in Sec. 6 we -1.5 0 200 400 600 800 1000 1200 discuss and summarize the main conclusions of the t paper. Fig. 1. Time series for the action potential for the Hindmarsh–Rose model for r =0.003 and different values of 2. Complex Bifurcation Structures the current applied: (a) I =1.26, (b) I =1.28, (c) I =1.67, (d) I =3.20, (e) I =3.29, (f) I =3.34, and (g) I =3.50. The Hindmarsh–Rose model describes the dynamics of the membrane potential in the axon of a neuron, x(t), by means of a three-dimensional system oscillatory solutions than can be simple periodic fir- of nonlinear first-order differential equations, which ings of a single spike [Figs. 1(b) and 1(g)], periodic written in dimensionless form read: firings of well-defined bursts of spikes [Figs. 1(c) and 1(d)], or chaotic firings of spikes and bursts of x˙ = y +3x2 − x3 − z + I, (1) spikes [Figs. 1(f) and 1(e)]. Most of these dynam- Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com 2 ical behaviors being oscillatory, they can be char- y˙ =1− 5x − y, (2) acterized by means of the time intervals between 8 consecutive peaks, Ti (i =1, 2,...,N), for each z˙ = r 4 x + − z . (3) 5 function x(t). In neurobiology, these time intervals, Ti, are known as inter-spike intervals, and it is The other two dynamical variables, y(t)andz(t), believed that in the structure of inter-spike intervals describe the exchange of ions through the neuron is where neurons encode the information [Sejnowski,

by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. membrane by means of fast and slow ion channels, 1995; Abeles, 2004]. Therefore, they are the relevant respectively. The main parameters of the model are observables for neuronal dynamics; because of this, the current that enters the neuron, I,andtheeﬃ- they will be our basic observables here. ciency of the slow channels to exchange ions, r. We have used them to construct one- The dynamics that one can observe for the dimensional bifurcation diagrams as those pre- action potential, x(t), are quite diverse as illus- sented in Fig. 2. These have been obtained through trated in Fig. 1. There are equilibrium solutions the numerical integration of the equations of where the action potential decays to a constant motion, Eqs. (1)–(3), by means of a fourth-order value, as shown in Fig. 1(a), as well as a variety of Runge–Kutta algorithm with time step 0.005. The Complex Bifurcation Structures 3073

(a) (b) Fig. 2. Two examples of bifurcation diagrams computed for different bifurcation parameters: (a) for I changing and r fixed (r =0.003), and (b) for r changing and I fixed (I =3.25). A vertical dashed line signals the point where the continuous interior crisis occurs, and vertical dotted lines separate the blocks, identified by its periodicity number, p. For (b) only a selected set of blocks of low periodicity have been signaled to avoid having the figure cluttered up with dotted lines.

value of one of the system parameters, r or I,was bursts of spikes are separated by lapses of qui- fixed, and time series solutions were obtained for a escence [Fig. 1(e)]. The dynamics in the spiking set of different values of the other parameter, which regime is characterized by one time scale, that of is considered a bifurcation parameter. Then, plot- the spikes, while in the bursting regime there are ting as dots the different values of Ti obtained for two time scales, one for the bursts and other for the each of the values given to the bifurcation param- spikes. The transition between these two dynami- eter, the bifurcation diagrams were obtained. We cal regimes is what is called a continuous interior will note that the conclusions that follow would crisis. For a detailed qualitative and quantitative had been the same if more conventional observables study of the these crisis the reader is referred to to construct the bifurcation diagrams, such as the [González-Miranda, 2003]. maxima of x(t) or the times of crossing a surface in Block structured dynamics is observed to the phase space, had been used. left of the bifurcation diagram, for 1.28 ≤ I ≤ 3.31. Representative results of what is obtained for In this case the dynamical behaviors available can fixed r, and taking I as bifurcation parameter be classified in blocks accordingly to its period- appear in Fig. 2(a). In this case, the value of r is icity, p. Each block can be defined giving a seg- taken as 0.003, which is the same used to obtain ment of I where all motions have the same peri- the time series shown in Fig. 1. This bifurcation odicity, and whose limits are period-adding bifurca- Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com diagram provides an illustration of two potentially tions. For I<1.28 the dynamics falls to a fixed biologically significant bifurcation structures: con- point, being the interval for p =1verysmall tinuouscrises[González-Miranda, 2003], and block I ∈ [1.28, 1.30]. It deserves to be noted that the structured dynamics [González-Miranda, 2005]. dynamics within a block can be purely periodic, A continuous crises shows up as a sharp change as observed here for the blocks of low periodic- in the width of the bifurcation diagram at IC ≈ ity, or may have chaotic regions inside, as displayed 3.31. To the right of the crisis, we have the well- for the blocks of large periodicity. In these chaotic

by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. known bifurcation structure of an inverted period regions, the average values of the periodicity, as well doubling cascade. A continuous interior crisis is a as of other statistical measures of the distribution chaos–chaos transition between two qualitatively of the inter-spike intervals are similar to those of different dynamical behaviors in phase space. On the periodic part of the block where they belong. the side where the bifurcation diagram is nar- A detailed study of block structures dynamics can row we have spiking dynamics, characterized by be found in [González-Miranda, 2005]. Figures 1(a)– an irregular firing of spikes like the one shown in 1(g) further illustrate all this: in Fig. 1(a) we see the Fig. 1(f). In the wide part there is bursting dynam- decay to the equilibrium, in Figs. 1(b)–1(d) we have ics, which is a complex dynamical behavior where examples of periodic motions in blocks p =1, 3, 9, 3074 J. M. González-Miranda

respectively. Figure 1(e) shows the chaotic dynam- ranges. To consider the change of these parame- ics in the region of periodicity p =9,andFigs.1(f) ters is meaningful. To change I in a certain inter- and 1(g) the chaotic and period-1 dynamics after val means to change the intensity of the current the continuous crisis, respectively. injected in the neuron. The variation of r allows A bifurcation diagram when r is taken as the two interpretations. On one side, different values of bifurcation parameter is shown in Fig. 2(b) which r might refer to consider different neurons, having corresponds to I =3.25. In this case, a continu- each different densities of slow ion channels in their ous interior crisis, which is very sharp, occurs for axon membranes. To allow a relatively wide range small values of the bifurcation parameter. Period of variation for r is meaningful because neurons can adding bifurcations develop as r decreases and a be of many different types, and to present a great large number of different blocks are observed, while variability between individuals. A second reason for chaos occurs mainly within the blocks of low peri- changing r is to take into account the existence of odicity here. Although the same type of dynamic ionic channels that can be chemically activated or behaviors seen for fixed r also appear in this case, deactivated, so that the permeability of the axon they occur in a somewhat different way. membrane of a given neuron can be modified by means of the presence or absence of the appropriate activating–deactivating chemical. The variations of 3. Linear Stability Analysis r would then take into account this possibility. The results in the above section reveal a variety of We start our study of the two-dimensional dynamical behaviors in the Hindmarsh–Rose model: bifurcation diagram with a linear stability analy- there are equilibria and oscillations, the oscillations sis of Eqs. (1)–(3). They have a single fixed point can be of a variety of types depending on their peri- (i.e. equilibrium) which is determined by the only odicity, and there is chaos. It is then interesting to real root of the polynomial investigate how they are distributed along its two- r − I 3 27 dimensional space of parameters, .Mostof p(x)=x +2x +4x + − I , (5) the research reported on this model has been done 5 giving numerical values to I ≈ 3, and to r ≈ 0.006 which depends on I but not on r. The real root, or r ≈ 0.001, as can be seen, for example, in the xF (I), determines the other coordinates of the fixed papers by Dhamala et al. [2004], Huang [2004] and point by means of Percha et al. [2005], to name a few of the most 2 recent references on research where the Hindmarsh– yF (I)=1− 5xF (I), Rose model plays a relevant role. This is a limited 32 (6) set of parameter values, mainly if one takes into zF (I)=4xF (I)+ . account that Hindmarsh and Rose [1984] studied 5 the dynamics for four values for these parameters The stability of this equilibrium is given by the (including the above) that were scattered within the eigenvalues of the Jacobian matrix Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com ranges −3.0 ≤ I ≤ 4.0and0.001 ≤ r ≤ 0.005.   6x − 3x2 1 −1 In this paper, we report the results of a system-   atic study of the dynamical behaviors available to J =  −10x −10 (7) the Hindmarsh–Rose model in a rectangle 4r 0 −r R = {(r, I)|r ∈ [10−4, 0.05] and I ∈ [−8.0, 8.0]}, of the flow [Eqs. (1)–(3)], which are the roots of its (4) characteristic polynomial: −4 by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. whose edges are the segments [10 , 0.05] and 3 2 P (Λ) = Λ + a(r, I)Λ + b(r, I)Λ + c(r, I), (8) [−8.0, 8.0], for r and I, respectively. This is a region of the parameter space that, regarding the two being the coefficient functions of r and I: parameters, is larger than the region where Hind- 2 marsh and Rose [1984] performed their study. We a(r, I)=1+r − 6xF (I)+3xF (I), (9) 2 note, however, that the upper limit that we con- b(r, I)=[5− 6xF (I)+3xF (I)]r sider for r is still small enough so that Eq. (3) can 2 +4xF (I)+3x (I), (10) be a description of the dynamics of slow ion chan- F 2 nels, and the values for I are still within realistic c(r, I)=[4+4xF (I)+3xF (I)]r. (11) Complex Bifurcation Structures 3075

We have determined xF (I) ∈ R and Λ(r, I) ∈ C Regarding Λ(r, I), we have obtained a first as the roots of p(x)=0andP (Λ) = 0, respec- eigenvalue Λ1(r, I) which is real and negative for tively. These have been computed exactly as the all (r, I) ∈R, and two complex conjugate eigenval- solutions of these cubic equations obtained by radi- ues, Λ2(r, I)andΛ3(r, I), whose real and imaginary cals, using the Cardano–Vieta formulas [Birkhoff & parts contain the essential information to classify Mac Lane, 1996]. The results for xF (I), presented in the equilibrium. All eigenvalues found have no zero Fig. 3 show a monotonous increase of xF (I)withina real parts, so we conclude that the system is hyper- narrow range of values; while, zF (I), also increases, bolic in R; therefore, the nonlinear dynamics near although in wider range, and yF (I)isthevariable the equilibrium point resemble the linearized sys- which suffers a larger variation, this increases for tem [Hirsch et al., 2004]. The results for the eigen- I 5.0 and decreases slightly for I 5.0. values are summarized in Fig. 4 where the real and imaginary parts of Λ2(r, I) are displayed as 3D-color plots. The plot [Fig. 4(a)] of the real part of the second eigenvalue, Re[Λ2(r, I)], which is identical 9.0 to that of Re[Λ3(r, I)] (not displayed), shows that R is separated in four regions which have approxi- 0.0 mately the shape of horizontal strips which we will name from bottom to top as regions A, B, C and D A C x .Inthisfigure,regions and appear colored F -9.0 y violet-blue, and regions B and D orange-red. F I zF For increasing values of , they are as follows -18.0 (see Fig. 5). For negative and small positive values of I,wehaveregionA, which is characterized by Re[Λ2,3(r, I)] < 0. This region occurs for values of -27.0 I below a certain curve I = φA(r) that increases monotonously within the interval I ∈ [1.2, 2.0]. -8.0 -4.0 0.0 4.0 8.0 This line is visualized as the edge between the I lower blue and orange regions. In region B,itis 3 Fig. 3. The only real root, xF (I), of x +2x+4x+[(27/5)− Re[Λ2,3(r, I)] > 0 for a range of values of I that are I] = 0 (red line), and the other coordinates of the fixed above φA(r) and below and almost a horizontal line, X x ,y ,z point, F =( F F F ), of Eqs. (1)–(3), determined from I = φB(r), that occurs around I ≈ 5.2 (again as the y I z I it: F ( ) (green line), and (c) F ( ) (blue line). edge between orange and blue regions). Region C, Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

(a) (b) 3 2 Fig. 4. (a) The real, Re[Λ2(r, I)] and, (b) imaginary, Im[Λ2(r, I)], parts of the second root of Λ + a(r, I)Λ + b(r, I)Λ + c(r, I) = 0, as functions of r and I. The color code is: orange-red for positive, green for zero, and blue-violet for negative values. 3076 J. M. Gonz´alez-Miranda

8.0 Im[Λ2(r, I)] < 0 to the right; so to the left the equi- sSc(-) D librium is a sink and to the right it is a spiral sink. For the unstable region B it is also Im[Λ2(r, I)] = 0 sSn(+) sSn(-) C to the left, and Im[Λ2(r, I)] < 0totheright;soto Sd sSc(-) 4.0 B the left there is a saddle and to the right a spiral source. For the stable region C it is Im[Λ2(r, I)] > 0 to the left, and Im[Λ2(r, I)] < 0totheright;soin the two cases we have spiral sinks. Finally, the whole D Im r, I < I 0.0 sSn(-) region is characterized by [Λ2( )] 0, so the dynamics there is that of a spiral source. A

-4.0 4. Nonlinear Analysis Sn sSc: Spiral Source sSn: Spiral Sink More insight on the nature and stability of the Sd: Saddle dynamical behaviors available to the Hindmarsh– Sn: Sink Rose model has been obtained, going beyond linear -8.0 0 0.01 0.02 0.03 0.04 0.05 stability, by means of the calculation of the spec- r tra of Lyapunov exponents, λi(r, I)withi =1, 2, 3. Fig. 5. Two parameter (r − I) bifurcation diagram showing These have been obtained numerically using stan- the different stability regimes (labeled A, B, C, D), separated dard techniques for systems of differential equations by continuous lines, which represent the functions I = φi(r), [Wolf et al., 1985]. The initial condition for such i = A, B, C. The functions ψA(I)andψB (I) appear as dotted calculations were far from the equilibrium point, so lines. The dynamics observed in each region, are identified by that they describe the nonlinear dynamics of the means of abbreviations as mentioned in the figure legend, and system. for the case of spirals the sign of Im[Λ2] is given in parenthe- sis. The dashed line is the boundary of the enclosure, E,where The results for the largest Lyapunov exponent complex nonlinear bifurcation structures develop (Sec. 4). presented in Fig. 6 allow us to distinguish the different dynamical attractors available: an equilibrium when λ1(r, I) < 0, a cycle when λ1(r, I) = 0, and where again Re[Λ2,3(r, I)] < 0, is above φB(r)and chaos when λ1(r, I) > 0. A check of the sign of the below a third line, I = φC (r), that occurs around second Lyapunov exponent, λ2(r, I), allowed us to I ≈ 6.0. Finally, for I above φC (r)wehaveregion exclude the possibility of quasiperiodic motion for D,whereRe[Λ2,3(r, I)] > 0. Since Λ1(r, I) < 0 the λ1(r, I) = 0 cases. Because the Lyapunov expo- within all the rectangle R, we have that in regions A nents are computed numerically, we never obtain an and C the fixed point is a stable equilibrium, while exponent that is zero with infinite precision; there- in regions B and D this equilibrium is unstable. fore an exponent, λ,isconsideredtobezerowhen More details on the nature of the dynamics 0− <λ<0+,with0− and 0+ constant numbers Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com around the fixed point are inferred from the imag- with small absolute values given by the statistical inary part of the second eigenvalue, Im[Λ2(r, I)] = errors involved in the calculation which are of the −3 −Im[Λ3(r, I)]. This is presented in Fig. 4(b), which order of 10 . As shown in Fig. 6(a), for I below shows that, regarding the imaginary part of Λ2(r, I) the critical line I = φA(r) the dynamics falls to and Λ3(r, I), the above regions, A, B and C,are an stable fixed point, in agreement with the results separated in a left and a right zone by curves, of linear stability analyses. For I above φA(r), we r = ψa(I), defined for I ∈ (−4.8, 5.4), and r = obtain cycles and chaotic dynamics interwoven in

by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. ψb(I), defined for I ∈ (5.4, 6.1). The first of these complex structures, that occur mainly for small val- curves, which has the shape of an oscillation, sepa- ues of r and I>0, as shown in detail in Fig. 6(b). rates regions where Im[Λ2(r, I)] = 0 (green) from Chaotic dynamics, which occurs in region B,mainly regions where Im[Λ2(r, I)] < 0 (blue); whereas, appears in areas having the shape of vertical strips ψb(I), which is defined for larger values of I,sep- that merge on a line with small slope that goes from −4 −3 arates regions where Im[Λ2(r, I)] > 0(red)from (r, I) ≈ (10 , 3.1) to (r, I) ≈ (6 · 10 , 3.4) in the regions where Im[Λ2(r, I)] < 0 (blue). From the r − I plane. shape of ψa(I)andψb(I) we see that in the sta- The long-term linear and nonlinear dynamics ble region, A,itisIm[Λ2(r, I)] = 0 to the left, and do not match in region C, where there is bistability Complex Bifurcation Structures 3077

6.78 7.0 6.5 6.75 z 6.0 6.72 5.5 5.0 0.6 0.7 0.8 0.9 1.0 -6.0 -4.0 -2.0 0.0 y y 0.3 x 0.2 0.1 0.0 2.4 (a) x 1.2 0.0 -1.2 0 60 120 180 t

0.83 8.0 6.0 0.82 z 4.0 0.81 2.0 0.80 0.0 -8.8 -8.7 -8.6 -90.0 -60.0 -30.0 y y

Fig. 7. (a, b) Projections of trajectories onto the y −z plane of the phase space, and (c, d) time series in the bistable region C, for r =0.03, I =5.8 and initial conditions (b) (a, c) (0.3, 0.6, 6.7) and (b, d) (0.3, 0.6, 7.0). Projections of trajectories onto the y − z plane for r =0.03, I =1.0and Fig. 6. The largest Lyapunov exponent, λ1(r, I), for Eqs. initial conditions (e) (−1.4, −8.7, 0.8) and (f) (14.0, 87.0, 8.0). (1)–(3), as functions of r and I: (a) in the rectangle R,and The position of the ﬁxed point is indicated by a ﬁlled circle (b) in a smaller region, R1, where the dynamical behavior in the phase space projections of trajectories, and by a dot-

Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com is more complex. The following color code is used: orange- ted line in the time series plots. Filled triangles indicate the red for chaotic dynamics, green-yellow for limit cycles, and initial condition of each trajectory in phase space projections. violet-blue for ﬁxed points. The interval where an expo- −3 nent is considered to be zero is given by 0− = −10 and −3 0+ =10 . at XF =(0.095, 0.955, 6.781). Whereas for X0 = (0.3, 0.6, 7.0), the system is attracted to a limit associated to the initial conditions that can lead the cycle in phase space [Fig. 7(b)], regarding the action system to an equilibrium state or to periodic oscil- potential this means a time series made from a peri-

by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. lations. This is illustrated by means of two exam- odic firing of a pulse [Fig. 7(d)]. ples in Fig. 7. The bistable behavior is shown by This is contrary to what happens in the other means of the plots of two trajectories, computed linearly stable region (region A), where no mat- for r =0.03 and I =5.8, that were started at ter if the initial conditions are close or far from very close initial conditions, X0 =(x0,y0,z0). For the fixed point the dynamics tends asymptotically X0 =(0.3, 0.6, 6.7), the projection of the phase to the equilibria. This is illustrated in Figs. 7(e) space trajectories onto the y − z plane [Fig. 7(a)], and 7(f) by means of projections of phase space and the time series for x(t) [Fig. 7(c)] show an trajectories computed for r =0.03 and I =1.0 oscillatory decay towards the fixed point located and initiated at X0 =(−1.4, −8.7, 0.8) and at 3078 J. M. González-Miranda

X0 =(−14.0, −87.0, 8.0), respectively. Since the inter-spike intervals for each point of the parameter fixed point at XF =(−1.394, −8.721, 0.822) the two space. trajectories share the same fate. We display the results obtained in Fig. 8(a) by The dynamics in region D is quite simple: we means of a 3D-color plot of a grid of 240×240 points have a stable limit cycle and an unstable equilib- in R1. For both, period-1 orbits and fixed points, rium point; trajectories escaping from this drop into the width of the bifurcation diagram is constant and the cycle. equal to zero. This appears in Fig. 8(a) as an outer From these results, we see that it is region B,in region, displayed as a flat violet color, which sur- the parameter plane, where the dynamical behavior rounds an enclosure, we will call it E, where colors of the Hindmarsh–Rose model is more complex, and range from blue to red. The area E is the region then more interesting. More precisely, the rectangle of parameter space (Fig. 5) where the dynamics is −4 more complex because the periodicity of the orbits R1 = {(r, I)|r ∈ [10 , 0.04] and I ∈ [1.1, 3.7]}, is 2 or greater, or there is chaos. The limit of this (12) area is made by a line that is composed by a seg- contains the values of the parameters that lead ment of φA(r), in its lower part, and the curve where to complex periodic oscillations (i.e. bursts and period-1 cycles bifurcate to period-2 cycles. This spikes) and chaos. Then, we will concentrate now on line is clearly seen as the set of points where the this rectangle. The main feature of this complexity violet color of the outer region abruptly becomes is block structured dynamics [González-Miranda, blue or green for the period doubling bifurcation 2005], which can be characterized by the period- curve, or yellow-red for the curve φA(r). icity of the oscillations of x(t), and is suggested Inside E, the width of the bifurcation dia- by the strip-like appearance of the areas of chaotic grams increases, steadily between three and four motion. To explore this behavior, we have computed orders of magnitude when r decreases. When I a large number of one-dimensional bifurcation dia- is increased, it suffers comparatively smaller vari- grams like those in Fig. 2. To have a global view of ations, except in the lower and upper limits of all of them we define the width of the bifurcation E. In the lower limit, we have an abrupt change diagram at point (r, I)as along φA(r) which corresponds to a Hopf bifurcation where an equilibrium point exchanges stability W r, I T r, I − T r, I , ( )= M ( ) m( ) (13) with a limit cycle. In the upper limit we have with TM (r, I)andTm(r, I) the maximum and mini- the continuous interior crises previously described mum inter-spike interval observed for (r, I), respec- [González-Miranda, 2003]. These are seen as a well- tively. This is a measure of the dispersion of the defined line inside R, where a blue region suddenly Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

(a) (b)

Fig. 8. (a) The width, W (r, I), of the bifurcation diagram and (b) the modulus of its gradient S(r, I) in the rectangle R1. The intensity of the color scales provides us with measures of the corresponding magnitudes. Complex Bifurcation Structures 3079

becomes orange or light green. All this is in accor- regions have the shape of narrow strips inclined dance with the particular one-dimensional bifurca- at about 65◦, where the periodicity increases, from tion diagrams shown in Fig. 2. right to left, by means of period-adding bifurcations More insight on the system dynamics within E which change p in a unit when the line is crossed. can be obtained from the modulus of the gradient The width of the strip decreases as the periodic- of W (r, I), ity increases, and they accumulate to the left of I . the figure. For intensities above =31thecas- ∂W 2 ∂W 2 cade of period-adding bifurcations ends abruptly in S r, I . ( )= + (14) an interior crises. Below this intensity our study, ∂r I ∂I r which was done for r ≥ 10−4, shows no end to This is suggested by one-dimensional bifurcation this cascade. diagrams, like those in Fig. 2, which show what has This is further illustrated in Fig. 9 by means been called block-structured dynamics. The main of additional plots devised to enhance the visual- feature of these structures is that within a block ization of these issues. In Fig. 9(a) it appears the the dynamics has a well-defined periodicity. The derivative (∂W/∂I)r, as a function of (r, I) ∈R1. limits between blocks are characterized by what we This quantity has been chosen because plots of one- might call a fine structure in W (r, I): there are some dimensional bifurcation diagrams, such as those in abrupt tiny changes in the width of the bifurca- Fig. 2(a), where I is chosen as bifurcation param- tion diagram around each period-adding bifurca- eter suggest that the maxima that separate blocks tion. This is a feature, only suggested in the plot are sharper. This is presented as a 3D-color plot of W (r, I), that can be better seen from a glance in Fig. 9(a), where the scale of the z-axis has been at S(r, I), which is displayed in Fig. 8(b). W (r, I) adjusted to better tune this fine structure by giv- being constant outside E, the limits of the enclo- ing a black color to points where (∂W/∂I)r < 1. sure are also clearly seen in this plot; but, what This results in a series of blue-violet colored lines is more important, is that we can appreciate the within E that signal the limits between blocks. presence and extent of block structured dynamics. These are clearly seen up to p = 16. Because the The limits of the blocks are seen as yellow lines, width of the block decreases with p, the blocks with slopes around 65◦. Therefore, the interior of start to be indistinguishable for p>16 when its E can be divided in regions characterized by a width becomes commensurable with the resolution given periodicity, p ≥ 2, of the dynamics. These of the picture. This picture has been modified so Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

(a) (b) Fig. 9. (a) Derivative of the width of the bifurcation diagram with respect to I and r fixed (colored points) with the color adjusted to better display block structured dynamics. On top of this plot, as white dots, we have signaled the points where the first Lyapunov exponent is greater than zero. (b) The width, W (r, I), of the bifurcation diagram around the continuous interior crisis. 3080 J. M. González-Miranda

that the points, (r, I), where the neuron dynam- taken into consideration. As originally written by ics is chaotic have been painted white to show that Hindmarsh and Rose [1984], Eq. (3) reads the strip-shaped chaotic regions match the limits z˙ = r[s(x − δ) − z], (15) between blocks. Moreover, Fig. 9(a) also shows very clearly, as a red-orange line near the bottom of the where s and δ are additional parameters that graph where Hopf bifurcations turn the equilibria describe details of the ion transfer through slow to cycles. channels. Usually, these are held fixed at the values The line where the crisis occurs, near the top of s =4andδ = −8/5 given by Hindmarsh and Rose E, is better seen in Fig. 9(b), which shows a detail [1984], just as we have done here. Some authors (e.g. of Fig. 8(a) around the crisis region. The abrupt [He et al., 2001; Rosenblum & Pikovsky, 2004]), change in the size of the attractor (between one however, have considered δ = −1.56, and Hind- to four orders of magnitude) is seen as a sharp marsh and Rose [1984] have also given some limited ◦ line, with a slope of around 40 , that separates attention to s = 1. Therefore, to study the struc- blue-green from red-yellow regions. The upper blue- tural stability of the results reported in this arti- green region corresponds to the inverted period dou- cle, we have performed some additional calculations bling cascade that leads to period-1 cycles that of eigenvalues, Lyapunov exponent and bifurcation lie in the violet region. The lower orange-yellow- diagrams in the plane r − I for several values of δ green regions is where block structured dynamics and s. occurs. The crises are the transition points between Regarding linear stability analysis, fixed points these two qualitatively different nonlinear dynam- are now determined from the roots of p(x)= ics. The abruptness of the crises decreases steadily x3 +2x + sx + α,withα = −(1 + sδ + I)and −3 as r increases, and for r ≈ 3 · 10 we can hardly being, in this case, zF (I)=sxF (I) − sδ.The speak of such crises. coefficients b and c of Eq. (8) are now b(r, I)= 2 2 [1 + s − 6xF (I)+3xF (I)]r +4xF (I)+3xF (I), and 2 c(r, I)=[s +4xF (I)+3xF (I)]r. The corresponding 5. Structural Stability polynomial equations have been solved as in Sec. 3. Despite I and, sometimes, r are the control param- To test structural stability under changes of δ eters that are allowed to change in most of the we have computed the eigenvalues, Λ1,2,3(r, I), for published research involving the Hindmarsh–Rose δ = −1.4andδ = −1.8. In Fig. 10(a) we summa- model; there are two other parameters that can be rize in a single plot the results for Λ2(r, I)when Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles.

(a) (b) 3 2 Fig. 10. The imaginary part of the second root of Λ + a(r, I)Λ + b(r, I)Λ + c(r, I)=0,Im[Λ2(r, I)], plotted using the color code: red for Im[Λ] > 0, green for Im[Λ] = 0, and blue for Im[Λ] < 0. The white lines are the functions I = φi(r), i ∈{A, B, C} from bottom to top, which separate stable (A, C) and unstable (B, D) regions. The parameter values modiﬁed are (a) δ = −1.4, and (b) s =1.0. Complex Bifurcation Structures 3081

(a) (b) Fig. 11. Plots of S(r, I), the modulus of the gradient of the width of the bifurcation diagram, for Eqs. (1)–(3), in the regions where complex bifurcation structures develop for the parameter values: (a) δ = −1.4, and (b) s =1.0. As done in Fig. 9(a), the points where the ﬁrst Lyapunov exponent, λ1, is greater than zero are superimposed as white dots.

δ = −1.4, which are very much similar to those for and position of these curves have largely modified. δ = −8/5=−1.6,presentedinFig.4.Asinthat The imaginary part of Λ2, also presents a similar case, Λ1(r, I) has been found real and negative, and structure, except for the zone where it is zero which: Λ2(r, I)andΛ3(r, I) are complex conjugate eigen- (i) has been notably diminished in favor of the nega- values, which behave in essentially the same fashion. tive zone, to the point that it disappears in A and is Therefore, here we have also four alternated regions, much smaller in B, and (ii) has entered the lower left A–D, whose stability is given by the real part of Λ2, corner of C (we note that this last effect is also seen which have the form of horizontal strips, separated in Fig. 10(a), although in a much smaller scale). by curves I = φi(r), i ∈{A, B, C},withA and C as The study of the Lyapunov spectrum and the stable regions. As before, the nature of the dynam- two-dimensional bifurcation diagrams leads to the ics in each region changes, depending on whether same kind of conclusions for the nonlinear dynam- the left or right part of the strip is considered. In ics. The results for δ = −1.4, shown in Fig. 11(a) fact, we see nearly the same results as in Fig. 4, are also very much similar to those in Figs. 8 and 9. shifted and amount to ∆I ≈ 0.9downalongthe With the only exception of a small displacement of I axis. The results for δ = −1.8, not represented, the entire figure down along the I axis, we see the Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com would have looked practically the same but for a dis- same features that we founded for δ = −8/5, and in placement of the same amount, this time up along particular, the complex bifurcation structures that the I axis. are the signal of block structured dynamics and con- The effect of s was studied by giving values tinuous interior crisis. For s = 1, Fig. 11(b) shows to s =3, 2, 1. For changes not too large, the pic- the same qualitative behavior seen in Figs. 8 and 9, ture given is Fig. 4, does not suffer modifications although now they occur in a region of parameter much larger than those discussed in the previous values that is quite different.

by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. paragraph for the effect of changing δ.Inpartic- ular, plots for s = 3 would not differ from the plots in Fig. 4 more than in Fig. 10(a). For large 6. Discussion and Conclusions changes there are quantitative, but not essentially We have provided in this article a descriptive view qualitative changes as illustrated by the results for of the different dynamical behaviors available to s = 1 given in Fig. 10(b). Regarding stability we the Hindmarsh–Rose neuron model in a wide and still have the two stable regions (A, C)thatalter- meaningful region of its parameter space. We have nate with the unstable regions (B,D) separated by determined subregions where the dynamics falls to curves I = φi(r), i ∈{A, B, C}, although the width an equilibrium point, where there are simple limit 3082 J. M. González-Miranda

cycles, and where there is bistability. Moreover, working in the enclosure where block structured more importantly, we have delimited the area where dynamics occurs as suggested by González-Miranda the dynamics is very complex, displaying burst- [2005] who proposed several coding mechanisms ing periodic and chaotic firings of spikes. We have based on the fact that each block could be asso- shown that there is some systematics in this region, ciated to a unit of meaning, and messages could be which we describe as block structured dynamics, constructed by switching between these units. as well as the existence of sharp transitions from Moreover, background information is provided simple equilibria or periodic dynamics to this com- to deal with many significant problems of nonlinear plexity. It deserves to be noted that these complex dynamics and chaos. These range from the study bifurcation structures most neatly develop for val- of the synchronization dynamics of nonlinear oscil- ues of r (around 10−3–10−2)andI (around 1.0–4.0), lators when the coupled oscillators are different which are in the vicinity of the values where the [González-Miranda, 2004], to expand the range of Hindmarsh–Rose model was originally fitted; thus, applications of bifurcation theory of vector fields possibly, the most realistic. by using it for a deeper understanding of the vari- Moreover, in our study of structural stabil- ety of bifurcation scenarios described in this article ity, we have observed that these complex bifurca- [Guckenheimer and Holmes, 2002]. tion structures are also displayed by this model In conclusion, the phase diagram of the for other sets of parameter values. This is an indi- Hindmarsh–Rose model, studied in a significant and cation that such structures are intrinsic funda- realistic domain of its space of parameters, displays mental properties of the Hindmarsh–Rose model. a complex structure. The outer regions correspond Furthermore, it has been observed [González- to simple behaviors such as equilibria or limit cycles, Miranda, 2005] that one-dimensional bifurcation which might be interpreted as states where the neu- diagrams like those plotted in Fig. 2, displaying ron is at rest or standing by. Moreover, core regions block structured dynamics and continuous inte- exist where different oscillatory dynamical behav- rior crisis, have also been observed in other mod- iors can be classified according to their periodicity els of neurons such as the Chay [1985] model in blocks, which might be the dynamical elements and the modified Hodgkin–Huxley model of ther- implied in neuronal coding; therefore, these are the mally sensitive neurons [Brown et al., 1998]. working regions of neuronal activity. The transi- Although, no systematic studies like the one pre- tion between core regions and outer regions can be sented here have been performed in these cases, very sharp, allowing rapid responses to stimulus. We those restricted observations suggest that com- think that these results provide useful background plex bifurcation structures like the ones studied information to help the progress of the research on here might be an essential property of neuronal synchronization phenomena between coupled neu- dynamics. rons as well as in the search for the neuronal coding The results presented in this article provide mechanisms. information that is potentially useful for both Int. J. Bifurcation Chaos 2007.17:3071-3083. Downloaded from www.worldscientific.com improving our understanding of how neuronal sys- Acknowledgment tems work and to further develop the knowledge on the dynamics of nonlinear systems. Research supported by DGI through Grant No. For example, abrupt changes of the dynamics BFM2003-05106. such as the interior crisis and the Hopf bifurcations provide possible mechanisms to understand how a References nervous system can give rapid responses to stim- Abeles, M. [2004] “Time is precious,” Science 304, 523– by SOUTH CHINA UNIVERSITY OF TECHNOLOGY on 01/01/19. Re-use and distribution is strictly not permitted, except for Open Access articles. ulus [González-Miranda, 2003] because qualitative changes of the dynamics occur in response to a 524. Birkhoff, G. & Mac Lane, S. [1996] A Survey of Modern small system parameter or stimulus modification. Algebra, 5th edition (The Macmillan Company, NY). An alternative, or may be additional, mechanism Braun, H. A., Huber, M. T., Dewald, M., Schäfer,K.& for such quick responses is given by bistability, as Voigt, K. [1998] “Computer simulations of neuronal suggested by Foss and Milton [2000], which is a phe- signal transduction: The role of nonlinear dynamics nomenon that we have also observed here. Other- and noise,” Int. J. Bifurcation and Chaos 8, 881–889. wise, the neuronal coding of information needed to Chay, T. R. [1985] “Chaos in a three-variable model of perform complex tasks can be performed by neurons an excitable cell,” Physica D 16, 233–242. Complex Bifurcation Structures 3083

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